novel concepts in unmanned aircraft … virtual flight simulation using computational fluid dynamics...

Novel Concepts in Unmanned

Aircraft Aerodynamics, Flight

Stability, and Control

CONTENTS

Preface v

1 Virtual Flight Simulation using Computational Fluid Dynamics 1

1.1 Introduction 1

1.1.1 Flight Simulation 2

1.1.2 High–fidelity Analysis for Conceptual Aircraft Design 4

1.1.3 Aircraft Equations of Motion 5

1.1.4 Research Challenges 7

1.1.5 Aircraft Test Cases 9

1.2 Aerodynamic Model for Flight Simulation 11

1.2.1 Tabular Aerodynamic Database 11

1.2.2 Stability Derivatives Approach 13

1.2.3 Sources of Aerodynamic Predictions 15

1.3 Generation of Tabular Aerodynamic Model 19

1.3.1 Brute–force Approach 20

1.3.2 Surrogate Model 20

1.3.3 Adaptive Design of Experiment 27

1.3.4 Cognitive Sampling Algorithm 30

1.3.5 Data Fusion 32

1.3.6 Prediction of Dynamic Derivatives 35

1.4 Time Accurate CFD for Flight Simulation 37

1.4.1 Advanced Mathematical Models 38

References 45

PREFACE

1Virtual Flight Simulation using

Computational Fluid Dynamics

U. Akram 1 Lockheed Martin Commercial Flight Training, Sassenheim, 2171 AH, The

Netherlands

M. Cristofaro 2 AVL List GmbH, Alte Poststraße 152, 8020 Graz, Austria

A. Da Ronch 3 University of Southampton, Southampton, S017 1BJ, United Kingdom

The aim of this Chapter is to present recent advances in the development of high–fidelity

simulation software for virtual flight testing of unconventional and next–generation aircraft.

The accurate and efficient simulation of the full set of physics associated with aircraft flight

operations including flight control, aerodynamics, and structural dynamics is discussed.

Pivotal to this framework is a novel adaptive design of experiment algorithm for the rapid

generation of the aerodynamic database that is developed and demostrated on a complete

aircraft with highly nonlinear aerodynamics. Throughout the Chapter, several worked–out

examples and problems are introduced to aid the reader to gain competence and proficiency

in the methods and computer codes accompanying this Chapter.

The Chapter continues with Section 1.1 overviewing the state–of–the–art in virtual

flight simulation. It continues in Section 1.2 with a discussion of the aerodynamic model

traditionally used for flight simulation. Then, surrogate modelling and adaptive design of

experiment are developed in Section 1.3. Finally, advanced models used for virtual flight

simulation are presented in Section 1.4.

1.1 Introduction

Many aspects of conceptual through detailed stages of aircraft design demand accurate

understanding of the aircraft aerodynamics. Among other things, it is required to predict

the static and dynamic characteristics of the aircraft, assessing performance and flight

handling qualities, designing flight control laws and building aerodynamic models for flight

simulation.

1Aerospace Engineer, Performance Systems; [email protected] student, Development Engineer Software; [email protected] Frontiers Fellow, Lecturer of Aircraft Structural Design, Industry Secondee at Airbus Operations Ltd.;

[email protected]

This is a Book Title Name of the Author/Editor

c© XXXX John Wiley & Sons, Ltd

2 Virtual Flight Simulation using Computational Fluid Dynamics

Obtaining reliable aerodynamic information throughout the flight envelope at early stages

of the aircraft design process is a difficult task. This is not always due to the limitation

of the tools at the designer’s disposal but rather due to strict cost and time constraints

which encourage the use of the traditional design methods that are based on low–fidelity,

semiempirical and, mostly, linear formulations of aerodynamics originally developed for

conventional aircraft configurations. Though computationally efficient and relatively accurate

for flow regimes where the aerodynamics is dominantly linear, the traditional methods are

unreliable for flight regimes involving complex nonlinear aerodynamic phenomena resulting

from high angles of attack, increased rotational rates, flow separation, shock waves and their

mutual interaction. For unconventional aircraft configurations, nonlinear flow fields can be

encountered at low angles of attack in subsonic conditions which questions the applicability

of the traditional methods to these configurations even for benign flight regimes. This often

prohibits a detailed analysis of aircraft stability, performance and flight handling qualities

and camouflages possibly undesirable flight attributes during initial stages of the aircraft

design process. Discovering unwanted flight characteristics and control problems later in the

design process can lead to programme delays, costly redesigns or retrofittings and ultimately

degraded performances. There have been numerous examples of aircraft experiencing

uncommanded activity, as reported, for example, in ’Chambers and Hall (2004)’.

High–fidelity computational modelling and simulation has the capability for virtual wind

tunnel and flight testing. Flight simulation provides an alternative to traditional methods that

can enable the modern aircraft designer to scan potentially unlimited configurations. For

this approach to be successful, a high–fidelity aerodynamic model is imperative. However,

building a high–fidelity aerodynamic model requires high–fidelity simulations which are

expensive in terms of computational cost and time. This Chapter intends to introduce

computational methods which can provide reliable aerodynamic information throughout the

flight envelope and be cost and time efficient at the same time.

1.1.1 Flight Simulation

In the commercial aviation industry, flight simulation training devices are almost solely used

for training pilots and maintenance personnel. In the academia, they find their major use as a

research and teaching aid.

The scope of usage and performance of a flight simulation training device is determined

by the level of its simulation fidelity. In the field of simulation, fidelity can be translated

to realism, e.g. how close the simulation is to its real counterpart. A variety of flight

simulation training devices exist, ranging from relatively low fidelity desktop trainers to full

flight simulators which present the highest level of achievable fidelity, see Figure 1.1. A full

flight simulator normally consists of actual aircraft hardware, from cockpit panels and flight

controls to flight management computers among others. Fidelity is further strengthened by

a state of the art visual system linked to navigation databases, an extensive sound system

and a motion system which can simulate complex atmospheric effects such as turbulence and

buffet.

Not visible in Figure 1.1 is the software which is a major component of any flight

simulation training device. The software houses the systems’ simulation, ground handling

model and the flight dynamics calculations. The flight dynamics calculations are dependant

on information about aerodynamic forces and moments which are extracted from the

Virtual Flight Simulation using Computational Fluid Dynamics 3

(a)

(b)

(c) (d)

Figure 1.1 Examples of flight simulators; (a) B–737 Full Flight Simulator in a Take-off Scene

(courtesy "Lockheed Martin Commercial Flight Training B.V."); (b) A Full Flight Simulator on Motion

(courtesy "Lockheed Martin Commercial Flight Training B.V."); (c) FSC1000 Desktop Simulator

(from http://flightsimulatorcenter.com/;) and (d) A hobby Flight Simulator

http://flightsimulatorcenter.com/;


aerodynamic model. When building a commercial flight simulation training device, the

aerodynamic model is almost always supplied by the aircraft manufacturer. This aerodynamic

model is first produced during the initial design stages and then improved as the design

parameters are frozen and more information about the aircraft is made available. Finally, the

aerodynamic model is further improved using data from wind tunnel tests and flight tests.

This process is rather laborious and expensive.

For a flight simulation training device to be deemed fit for pilot training, it has to be

certified by the governing aviation authorities like the Federal Aviation Administration

(FAA 4) and European Aeronautical Safety Agency (EASA 5). The aviation authorities

conduct a series of objective and subjective tests on the flight simulation training device

before granting certification. Objective testing encompasses aircraft performance, stability

and flight handling qualities. The aircraft manufacturer gathers and provides reference data

to the flight simulation training device manufacturer for the objective tests in accordance with

the guidelines defined by the aviation authorities. The reference data can either be from flight

testing or from the aircraft manufacturer’s simulation. The flight simulation training device

is then run using the same aircraft configuration and pilot inputs as the reference data. For

each of the objective tests, the response of the flight simulation training device is required to

lie within predefined tolerances of the reference data. If the response of the flight simulation

training device does not fall within the tolerances, it will be deemed unfit for pilot training.

The aerodynamic model, therefore, is critical to the fidelity of any flight simulation training

device.

1.1.2 High–fidelity Analysis for Conceptual Aircraft Design

Assessing performance, stability and flight handling qualities throughout the flight envelope

during the conceptual design stages is not always possible. Often, this is a direct result of

using traditional, low–fidelity computational methods and stand alone simulations, e.g. 2–

3 degrees of freedom (DoF) or uncoupled calculations to minimise the cost and time. It

is also important to mention that the traditional design methods that rely on semiempirical

formulations were developed for conventional aircraft configurations. As the industry pushes

for unconventional designs, the applicability of these methods becomes questionable even

for benign flight conditions.

Flight simulation has the capability to facilitate a better conceptual design process. It can

allow the designer to scan any aircraft configuration with a 6 DoF model. This can improve

the estimates otherwise obtained by semiempirical methods, allowing better understanding

of any coupled effects and enabling to fly specific flight scenarios such as those required for

aircraft certification. Further, if such a set–up is to be better than the traditional approaches,

it should be able to predict the performance, stability and flight handling qualities throughout

the flight envelope.

Fortunately, this can be achieved without the need of a high level flight simulation training

device. An integrated desktop simulation and test environment is viable for the initial design

stages. This has been successfully demonstrated in ’Scharl et al. (2000)’. However, this

requires high–fidelity mass properties information, engine model, ground handling model

and an aerodynamic database. The intention of this Chapter is to introduce a cost and

4http://www.faa.gov/, retrieved August 20, 2015.5https://www.easa.europa.eu/, retrieved August 20, 2015.

http://www.faa.gov/

https://www.easa.europa.eu/


time efficient methodology to generate aerodynamic data for total flight envelope without

compromising the accuracy of predictions.

It is important to note that a comprehensive software tool (CEASIOM) has

been developed to address these needs. The reader may find more information

at http://www.ceasiom.com.

1.1.3 Aircraft Equations of Motion

The equations of motion of an aeroplane are the foundation on which the framework

for flight dynamics studies is built. They provide the key to a proper understanding of

flying and handling qualities. At their simplest, the equations of motion can describe

small–perturbation motion about trim only. At their most complex, they can be completely

descriptive, simultaneously embodying static stability, dynamic stability, aeroelastic effects,

atmospheric disturbances, and control system dynamics for a given aeroplane configuration.

The equations of motion enable the rather intangible description of flying and handling

qualities to be related to quantifiable stability and control parameters, which in turn may

be related to identifiable aerodynamic characteristics of the airframe. For initial studies the

theory of small perturbations is applied to the equations to facilitate their analytical solution

and to enhance their functional visibility. However, for more advanced applications, which

are beyond the scope of this Chapter, the fully descriptive nonlinear form of the equations

might be retained. In this case, the equations are difficult to solve analytically and computer

simulation techniques become necessary to obtain a numerical solution.

Systems of Axes and Notations

The aircraft equations of motion are generally derived with respect to an inertial reference

frame. In flight dynamics, the earth–fixed reference frame is considered, subject to a few

assumptions, as the inertial reference frame. In the earth–fixed reference frame, the XE–

axis points towards North, YE–axis towards East, and ZE–axis is directed downwards (see

Figure 1.2).

The equations of motion may be referred to the aircraft–fixed coordinate system. It is

convenient to take the aircraft centre of mass as the origin of this coordinate system. Two

sets of aircraft–fixed coordinate systems are defined. The first is a pure translation of the

earth–fixed reference frame to the aircraft centre of mass, with axes denoted by XAE , YAE ,

and ZAE (see Figure 1.2). This allows defining the position of the aircraft in terms of X , Y ,

and Z coordinates. For the second coordinate system, the XAB–axis lies in the aircraft’s

plane of symmetry and points forward, the YAB–axis is perpendicular to the aircraft’s

plane of symmetry and is directed out towards the right wing, and the ZAB–axis lies in

the aircraft plane of symmetry and points vertically downwards. This second coordinate

system is commonly referred to as the body–fixed coordinate system and allows defining the

orientation of the aircraft. Based on the chosen orientation of the X–axis, different body–fixed

coordinate systems can be defined.

http://www.ceasiom.com


Figure 1.2 Systems of axes and notations

Governing Equations

The derivation of the aircraft equations of motion is detailed in any book on flight dynamics,

see for example ’Napolitano (2011)’. The conservation law of linear momentum yields the

three Equations 1.1.

FX = m (u + q w − r v)FY = m (v + r u − pw)FZ = m (w + p v − q u)

(1.1)

and the conservation law of angular momentum yields the three Equations 1.2.

L = m [ Ixx p − Ixy q − Ixz r+q (−Ixz p − Iyz q − Izz r) − r (−Ixz p + Iyy q − Iyz r) ]

M = m [ −Ixy p + Iyy q − Iyz r−p (−Ixz p − Iyz q − Izz r) + r (Ixx p − Ixy q − Ixz r) ]

N = m [ −Ixz p − Iyz q + Izz r+p (−Ixy p + Iyy q − Iyz r) − q (Ixx p − Ixy q − Ixz r) ]

(1.2)

The left hand side of the above equations contains the external forces (FX , FY , and FZ )

and moments (L, M , and N ) acting on the aircraft, and the right hand side describes the

aircraft dynamics. It is of interest to observe that the external forces and moments may include

contributions from the aerodynamics, thrust, ground operations, and aircraft weight. This

Chapter, in particular, focuses on the computation of the aerodynamic loads using accurate

yet rapid methods. However, before commencing the main task of this Chapter, an overview

of the underlying difficulties and research challenges is given.


1.1.4 Research Challenges

The impact of aviation on the environment is under scrutiny as never before, and both the

European Union and the U.S. have imposed strict targets on noise and emission pollution.

These targets, however, are unlikely to be achieved by aircraft designed with current industrial

design procedures, and considerable technological advances are desperately needed. To

make progress in this direction, routine use of high–fidelity coupled analysis is instrumental

to improve realism and confidence of simulations. This allows simulating critical flight

conditions, identifying undesired characteristics, and optimising the overall performances

well before the first prototype is built.

Based on the Authors experience in the field of virtual flight simulation, three technical

challenges should be addressed.

1. Efficient generation of the aerodynamic database for flight simulation.

2. Integration of high–fidelity computational models early in the aircraft design process.

3. Data fusion of various aerodynamic sources to increase the realism and fidelity of the

aerodynamic database.

Current state–of–the–art methods and tools are discussed in light of the above three technical

challenges. Advances beyond the state–of–the–art form the core of this Chapter, and will be

thoroughly discussed in the remaining Sections.

Efficient generation of the aerodynamic database

To generate the aerodynamic database of forces and moments for the expected flight

envelope, a large number of flow conditions must be calculated. Considering that the total

number of flight conditions can easily be in excess of O(106), the brute force approach

of computing every entry becomes intractable using CFD as the source of the data. The

issue of how to exploit the benefits of using CFD to improve aircraft design has been

the topic of a large body of work ’Da Ronch et al. (2011a); Mason et al. (1998); Snyder

(1990)’. The references cited above exemplify the need for improvements in computational

efficiency. Access to high performance computing (HPC) facilities is essential for numerous

examples of intensive CFD simulations, but to make progress in routinely using CFD

previous research has been concentrated on the development of computationally efficient

predictive aerodynamic models combined with CFD simulations.

Despite the large body of work in this area, the formulation of an efficient, automated, and

robust method for the identification of aerodynamic nonlinearities has proven elusive. Most

often, a large number of calculations is needed to ensure a good accuracy of the surrogate

model. The slow convergence of the surrogate model may be attributed to the non–optimal

distribution of sample points over the flight envelope. As a result, CFD calculations are

performed away from the areas where aerodynamic nonlinearities occur, missing critical

features that may impact negatively on the following design. The high nonlinearity of the

aerodynamic forces trends for a two dimensional case is presented in Figure 1.3. It illustrates

the lift coefficient map for a domain based on the angle of attack and Mach number for a

NACA 0012 aerofoil.

This Chapter discusses the formulation, development and implementation of a novel

method to increase iteratively the accuracy of the surrogate model. The adaptive design


of experiment algorithm has been found adequate to capture local maxima and minima,

improving over state–of–the–art methods, and satifying the stringent constraint set on the

maximum number of CFD calculations allowed within a given time frame and without the

use of exclusive HPC. This is presented in Section 1.2.1.

Figure 1.3 Lift coefficient dependency on angle of attack and Mach number for a NACA 0012

aerofoil

High–fidelity computational models early in the aircraft design process

Determining the stability and control characteristics of aircraft at the edge of the flight

envelope is one of the most difficult and expensive aspects of the aircraft development

process. Nonlinearities and unsteadiness in the flow are associated with shock waves,

separation, vortices and their mutual interaction, which can lead to uncommanded motion

and uncontrollable departure. If these issues are discovered at flight test, the aircraft

development can suffer significant delays, a rise in production costs and detrimental effects

on performance. There have been numerous examples of aircraft experiencing uncommanded

activity, as reported, for instance, in ’Chambers and Hall (2004)’. To provide a better

fundamental understanding of the flow physics which might lead to degraded characteristics,

computational approaches have been used ’Woodson et al. (2005)’. The development of

a reliable computational tool would allow the designer to screen different configurations

prior to building the first prototype, reducing overall costs and limiting risks ’Forsythe et al.

(2006)’.

Probably, the most important drawback is that the aerodynamicist in conceptual design

needs to project the potential performance of the design after a complete detailed

aerodynamic design has been done. A CFD analysis of a shape that has not been designed

is of no particular value. Indeed, a detailed wing design may take months to perform. The

second drawback is the cycle time for CFD analysis. Despite the advent of more powerful


HPC clusters, CFD analyses represent a bottleneck in the conceptual design process, which

requires that several configurations can be evaluated daily.

This Chapter demonstrated that the routine use of CFD for conceptual design of complete

aircraft is possible at a manageble cost. A key issue is a robust mesh generation tool that, in

this case, relayes on the open software SUMO 6. SUMO is a graphical tool for a rapid aircraft

geometry creation coupled to high efficient unstructured surface and volume grid generators.

It takes as input a very few basic geometry parameterization values and uses it to produce

surface and volume grids for nonviscous CFD simulations.

Data fusion of various aerodynamic sources

As the design process evolves from the preliminary to the conceptual phase, the aircraft

geometry is tentively frozen and experimental testing is started to corroborate and verify

the aerodynamic predictions. With more detailed aerodynamic data being generated, it is

instrumental to incorporate the new information into the existing aerodynamic database.

Hence, the fidelity of the database is iteratively enhanced during the aircraft design process.

This approach, called data fusion, combines aerodynamic predictions from different sources

to obtain one single database that is more accurate than each single database taken separately.

Two aerodynamic databases are incorporated or fused at a time. Generally, it is assumed

that one database is of low–fidelity/cost and the other one is of high–fidelity/cost. The

cheap evaluations of the low–fidelity database provide information on the trend of the target

function (qualitative behaviour), and expensive evaluations of the high–fidelity database

correct this trend with quantitative information. Because of costs, the number of cheap

evaluations is significantly larger than that of the expensive evaluations.

In practice, it may not be possible to set a clear separation between low– and high–fidelity

aerodynamic databases. For example the data obtained from an expensive wind tunnel testing

campaign might carry errors due to equipment limitations or external interferences. Although

the data may be generally not sensibly affected, the flow field is much more sensible close

to the aerodynamic nonlinearities appearance. In this case, an automated data fusion method

seems not to exist to date, and engineering experience proves still the best judgment.

1.1.5 Aircraft Test Cases

The methods described in this Chapter were developed to address the need for a robust and

systematic conceptual design tool for unconventional or next–generation aircraft. Of special

interest is the design of next–generation aircraft that depart significantly from previous

configurations. Existing conceptual design tools, which often rely heavily on correlations

and fitted historical data, do not provide the flexibility or sufficiently general performance

prediction capability to address arbitrary new designs.

Two test cases of this Chapter are for the Stability And Control Configuration (SACCON)

and the Transonic Cruiser (TCR) wind tunnel models, shown in Figure 1.4.

The SACCON model, see ’Vallespin et al. (2011)’, is representative of an Unmanned

Combat Air Vehicles (UCAV) and consists of a lambda wing with a sweep angle of 53

deg and a wing washout of 5 deg, Figures 1.4(a) and 1.4(b). The flow behaviour around

this UCAV is characterised by vortical flows. A dual vortex structure is apparent up

6http://larosterna.com/sumo.html [retrieved August 20, 2015].

http://larosterna.com/sumo.html


to about 17 deg angle of attack, when the dual vortices merge together causing strong

nonlinear aerodynamic performances. The experimental data from the SACCON wind tunnel

tests obtained in the 3.25 m×2.8 m NWB wind tunnel for the rounded leading edge

model, ’Loeser et al. (2010)’, are used in this Chapter.

The design of the TCR model, Figures 1.4(c) and 1.4(d), was made during the SimSAC

(Simulating Aircraft Stability and Control Characteristics for Use in Conceptual Design)

project 7. The final configuration includes an all–moving canard for longitudinal control.

More details on the model design are given, for example, in’ Rizzi et al. (2011)’. The aircraft

design is driven by the requirement for a design cruise speed in the sonic speed range.

The specification for a cruise Mach number of 0.97 was set to stress the shortcomings of

engineering methods traditionally used in the early design phase. A wind tunnel model was

built and wind–tunnel testing for static and dynamic conditions was performed in the wind

tunnel facilities at the Central Aerohydrodynamic Institute (TsAGI), ’Mialon et al. (2010)’.

(a) SACCON configuration (b) SACCON CFD analysis

(c) TCR configuration (d) TCR CFD analysis

Figure 1.4 The aircraft test cases of this Chapter: (a)–(b) SACCON UCAV configura-

tion, ’Vallespin et al. (2011)’; and (c)–(d) TCR configuration, ’Rizzi et al. (2011)’

7SimSAC project website: http://www.simsacdesign.eu [retrieved August 20, 2015]

http://www.simsacdesign.eu


1.2 Aerodynamic Model for Flight Simulation

For flight simulation, a model of the aerodynamic forces and moments is required. The

choice of an appropriate model is not unique, and may result from a compromise between

the available information at the time and the turnaround time of simulations. In general,

higher fidelity models of the aerodynamics are also more computationally expensive and

require a complex geometry description. These contrasting aspects between accuracy, cost

and time limit generally the complexity of models of the aerodynamics to simple relations,

at least in the early phases of the aircraft development process and for multi–disciplinary

optimisation analyses. Today, the most common model of the aerodynamics still relies on

a model developed in the 1910s with minor modifications. Aircraft performances, however,

have evolved considerably over the past 100 or so years, and the limitations of the traditional

model are now becoming more and more apparent. The aerodynamic properties of an

aircraft vary considerably over the flight envelope, and their mathematical descriptions

are approximations at best. The limit of the approximations is determined either by the

ability of mathematics to describe the physical phenomena involved or by the acceptable

complexity of the description. The aim is to obtain the simplest approximation consistent

with adequate physical representation. In the first instance, this aim is best met when the

motion of interest is constrained to small perturbations about a steady flight condition,

which is usually, but not necessarily, trimmed equilibrium. This means that the aerodynamic

characteristics can be approximated by linearising about the chosen flight condition. Simple

approximate mathematical descriptions of aerodynamic stability and control derivatives then

follow relatively easily. This is the approach pioneered by ’Bryan (1911)’, and it usually

works extremely well provided the limitations of the model are recognised from the outset.

But real flight does not obey to mathematical limitations.

1.2.1 Tabular Aerodynamic Database

Modeling the aircraft aerodynamics raises the fundamental question of what the

mathematical structure of the model should be. The functional dependencies of the force

and moment coefficients are in general complex, as they depend nonlinearly on present and

past values of several quantities, such as airspeed, angles of incidence, etc. The flow is often

considered quasi–steady, which presumes that the flow reaches a steady state instantaneously

and the dependence on the history of the motion variables can be neglected. One exception

to this assumption is the retention of the reduced frequency effects. With these underlying

hypotheses, the characterization of the functional dependencies is broken down as:

Ci = f1 (α, β,M, δ) + f2 (Re) + f3

(

Ω c

2U∞

)

+ f4

(

ω c

2U∞

)

(1.3)

for i = L,D,m, Y, l and n

The first term on the right hand side can be obtained in steady–state analyses and static

wind tunnel tests, the second term represents Reynolds number corrections and the last

two terms are measured from rotary balance and forced oscillation tests, respectively. The

decomposition is valid when the effects are separable and the superposition principle is valid.

The effects of rotary and forced oscillation are typically modeled as a function of the body


Table 1.1 Aerodynamic database format; "x" indicates a column vector of non–zero

elements

α M β δele δrud δail . . . p q r CL CD Cm CY Cl Cn

x x x - - - - - - - x x x x x x

x x - x - - - - - - x x x x x x

x x - - x - - - - - x x x x x x

x x - - - x - - - - x x x x x x

x x - - - - x - - - x x x x x x

x x - - - - - x - - x x x x x x

x x - - - - - - x - x x x x x x

x x - - - - - - - x x x x x x x

axis angular rates, angles of incidence and their first time derivatives. These derivatives were

introduced to obtain a closer correlation between predicted and observed aircraft longitudinal

motion, and for a conventional aircraft they represent the finite time that aerodynamic loads

at the tail lag the changes in downwash convected downstream from the wing.

Traditionally, data obtained from extensive wind tunnel and flight test campaigns are

tabular in form, expressing the dependencies of the aerodynamic loads on the flight and

control settings. This form represents also the standard format used in flight simulators,

for stability and control assessment, and for flight control system design and synthesis. As

illustrated in Table 1.1, forces and moments are tabulated as functions of the aircraft states

and control settings. Aerodynamic coefficients are in wind axes, and the aircraft states feature

the angles of incidence and sideslip, α and β, the Mach number,M , and the body axis angular

rates, p, q and r. All required control deflections are also included. Several assumptions

have led to the formulation used, limiting its validity when confronted with uncommanded

departures involving aerodynamic and aircraft motion cross–coupling.

Typical Size of the Tabular Model

In the general case, the six aerodynamic coefficients would be functions of all input variables,

resulting in a very large table (see Section 1.3). This is not normally necessary, and a less

coupled formulation of the aerodynamic coefficients is used instead. Each aerodynamic term

is formulated as dependent on three input variables. The main aerodynamic variables are

taken to be the angle of attack, α, and Mach number, M . Forces and moments are assumed

to depend on these variables in combination with each of the remaining variables separately.

The complete aerodynamic database is then divided into three-parameter sub-tables. Let nx

denote the number of values for the parameter x in the table, and let Nc denote the number

of aircraft control effectors. The dimension of the complete database, ndb, is as follows

ndb = nα · nM ·

(

nβ +

Nc∑

i=1

nδi +

3∑

i=1

nωi

)

(1.4)

where ωi indicates the body axis angular rates.


For the same example illustrated above, the total number of table entries would be less

than two–hundred. However, a reasonable aerodynamic database to cover the expected flight

envelope can easily require one hundred–thousand entries.

1.2.2 Stability Derivatives Approach

The functions f1, f2, f3 and f4 showed in Eq. (1.3), represent the dependency between

aerodynamic forces and moments coefficients and states and command variables. These

are the integral result of the pressure distribution over the aircraft surface for every state

condition. The stability derivatives approach linearize these functions, so that the motion of

the aircraft can be more readily analyzed. These derivatives are measures of how forces and

moments change as states and command variables change.

Start considering only the steady part of Eq. (1.3), f1. Variations in forces and moments

with reference to an initial condition of equilibrium, for a certain Mach number, can be

expanded using Taylor series and expressed as Eq. (1.5).

Ci = Ci0 +∂Ci

∂α∆α+

1

2

∂2Ci

∂α2∆α2 + . . .

+∂Ci

∂β∆β +

1

2

∂2Ci

∂β2∆β2 +

∂Ci

∂α

∂Ci

∂β∆α∆β + . . .

+∂Ci

∂δ∆δ +

∂2Ci

∂δ2∆δ2 +

∂Ci

∂α

∂Ci

∂δ∆α∆δ +

∂Ci

∂β

∂Ci

∂δ∆β∆δ + . . .

(1.5)

Neglecting higher order terms, the equation is linearized as follows:

Ci = Ci0 +∂Ci

∂α∆α+

∂Ci

∂β∆β +

∂Ci

∂δ∆δ

= Ci0 + Ciα∆α+ Ciβ∆β + Ciδ∆δ

(1.6)

Ciα is the f1 gradient term in the α direction and is one of the so called aerodynamic

derivatives. Based on whether a derivative reflects a change in force or moment based on

an aircraft state or a control surface deflection, they are referred to as stability and control

derivatives, respectively.

Static Aerodynamic Derivatives

Whereas the above linear superposition of the effects is considered adequate for most of

conventional aircraft configurations, unconventional geometries exhibit highly nonlinear

dependences on control deflections even at benign flow conditions for their particular

configuration layout. As an example, consider the SACCON test case (Figure 1.4) that has

been the subject of extensive numerical investigations and wind tunnel testing. Both the

numerical predictions and wind tunnel measurements have indicated the development of

a complex flow field which results in the principle of superposition being invalid. More

accurate and nonlinear models of the aerodynamics are therefore sought after to improve

the simulation of the next generation of aircraft.


Equation (1.5) also assumes that the aerodynamic forces and moments are an instantaneous

function of the aircraft states and control variables. For flight manoeuvres involving slow

angular rates and linear angle of attack regimes, the flow adapts relatively quickly and can

be reasonably assumed to be independent of time ’Vallespin et al. (2010)’. However, the time

invariance assumption is not true for manoeuvres involving high angular rates and angles of

attack where nonlinear and unsteady aerodynamic effects are dominant.

Consider Figure 1.5 representing the flow field around SACCON during a pull up

manoeuvre, designed to take the aircraft through the nonlinear angle of attack regime in a

short period of time: the angle of attack changes from -5 to 30 degrees in about 1 second.

The two different flow fields are from (Figure 1.5(a)) steady and (Figure 1.5(b)) an unsteady

time-accurate CFD calculation, respectively. The surface topology of SACCON results in

vortex formation at the leading edges, indicated by the darker blue regions. For the steady

case, the vortices are stronger and break down further downstream relative to the unsteady

time-accurate case. During such manoeuvres, the flow characteristics are actually a function

of the previous motion history of the aircraft. This is known as aerodynamic hysteresis,

i.e. the aerodynamics becomes time dependent. This time variant phenomenon has been

extensively investigated for aerofoils, wings and for full aircraft configurations ’Mueller

(1985); Venkatakrishnan et al. (2006)’. In this nonlinear flight regime, the model predictions

have been shown to degrade ’McCracken et al. (2012); Vallespin (July 2012)’ and provide

erroneous indications on the aircraft behaviour.

(a) Steady CFD solution (b) Unsteady time-accurate CFD solution

Figure 1.5 SACCON Surface Pressure Distribution During a Pull Up Manoeuvre, M = 0.1026 and

α = 26.3 , ’Vallespin et al. (2010)’

Dynamic Aerodynamic Derivatives

The aerodynamic predictions obtained via stability derivatives approach for nonlinear and

unsteady flight regimes can be improved by adding the motion rates terms f3 and f4 of

Eq. (1.3). These additions, referred to as dynamic derivatives, were introduced to obtain a


close correlation between predicted and observed aircraft longitudinal motion ’Greenberg

(1951)’.

For each aerodynamic force and moment, as we did for the steady flight condition, we

can retain the linear terms which are most relevant and de–couple their combined influence

into individual contributions. Most aircraft are symmetric and this property can be used to

our advantage to neglect the dependence of longitudinal forces and moments on lateral–

directional variables and vice versa. While the dependence on β is typically neglected for

a quasi–steady flow, the inclusion of the α term leads to an identifiability problem when

estimating the α and q derivatives ’DaRonch (July 2012)’. To avoid this problem, the two

terms are lumped together and an equivalent derivative is defined as Ciq = Ciα + Ciq for i= L, D and m.

Following from the discussion above, we present a nonlinear mathematical formulation

for the longitudinal and lateral aerodynamic forces and moments in quasi–steady flow as

presented in Eq. (1.7).

Ci = Ci0(α,M, β) + Ciq(α,M, q)cq

2V+ Ciδ(α,M, δ)δ

Cj = Cj0(α,M, β) + Cjp(α,M, p)bp

2V+ Cjr(α,M, r)

br

2V+ Cjδ(α,M, δ)δ

(1.7)

where i = L, D and m and j = Y , l and n.

The static aerodynamic contributions and the dynamic derivatives can either be determined

through measured or computed aerodynamic data. A discussion of the available methods is

presented in 1.2.3, whereas the dynamic derivatives are discussed in Section 1.3.6.

1.2.3 Sources of Aerodynamic Predictions

A prerequisite for realistic predictions of the stability and control characteristics of an

aircraft is the availability of a complete and accurate aerodynamic database. The choice

of which aerodynamic model to use balances cost and fidelity. The higher the fidelity of

the aerodynamic model to be used, the higher the execution time is normally. In the early

phases of aircraft development, the geometry is defined with limited fidelity which might

render expensive methods of limited use. A number of models are used and these are now

summarized.

Figure 1.6 shows knowledge available and resources committed at various stages of the

aircraft design process. The methods developed in this Chapter aim at bringing higher–

fidelity tools, that are generally used only in the later stages of the development process, early

in the design cycle. The benefits of doing so are to develop significant insights of increased

realism on the complex nonlinear interactions that may jeopardise the aircraft performances,

and to improve the aircraft design well before it becomes economically unfeasible.

Semi–Empirical Methods

Semi–empirical methods, which are the traditional engineering tool for conceptual design,

often rely heavily on correlations and fitted historical data, and do not provide the flexibility

or sufficiently general performance prediction capability to address arbitrary new designs.

The data compendium (DATCOM), for example, is a document of more than 1,500 pages


Figure 1.6 Aerodynamic model increasing fidelity and cost, adopted during the aircraft design

process steps

covering detailed methodologies for determining stability and control characteristics of

conventional "wing–fuselage" aircraft configurations. In 1979, it was programmed in Fortran

and renamed the USAF stability and control digital DATCOM, ’Williams and Vukelich

(1979)’. Digital DATCOM is a semiempirical method which can rapidly produce the

aerodynamic derivatives based on geometry details and flight conditions. This code was

primarily developed to estimate aerodynamic derivatives of conventional configurations,

and to provide all the individual component contributions and the aircraft forces and

moments. A design uncertainty factor is often needed to account for validity of aerodynamic

characteristics estimated using this method. Figure 1.7 shows the level of simplification of

the geometry used by DATCOM.

Numerical Methods

CFD provides a range of methods with varying fidelity which can be employed for

aerodynamic analyses of aircraft. The Navier–Stokes (NS) equations that describe the

motion of a viscous and compressible flow form the basis of any CFD analysis. The

underlying nonlinear partial differential equations have to be solved numerically with

appropriate algorithms, and provide the highest level of fidelity. NS computations are also


Figure 1.7 Geometry representation of a traditional aircraft in DATCOM

the most computationally expensive. Reduced fidelity equations can be achieved through

simplification of the NS equations. On the other end of the fidelity spectrum, the linear

potential flow theory is the most computationally efficient but its generality is strongly limited

to linear cases.

Full aircraft configurations can be analysed using high–fidelity CFD, ’McDaniel et al.

(2010)’. This can provide a better understanding of flow physics throughout the flight

envelope and this has been successfully demonstrated in the literature, ’Woodson et al.

(2005)’ and ’Ghoreyshi et al. (2010)’. However, the computing cost and time required for

high–fidelity CFD analysis of full aircraft configurations across their flight envelope limits

its use in the initial design process.

The error between low and high–fidelity CFD computations for benign flight conditions

is usually negligible. However for flight conditions where nonlinear flow characteristics

are dominant, this error increases considerably. Therefore, a combination of low and high–

fidelity CFD analyses enables the cost and time efficient generation of aerodynamic database

for aircraft. The low–fidelity analyses are used for benign regions of the flight envelope, and

high–fidelity analyses used for extreme flight conditions. Figure 1.4(b) shows the surface

pressure coefficient around the SACCON configuration at two angles of attack. The resulting

aerodynamic forces and moments are integrated from the surface pressure distribution.

Wind Tunnel Testing

Before the Wright brothers managed to take their Flyer I off the ground more than 100

years ago, they had performed a series of careful experiments with different wing models

in a small wind tunnel installed in their bicycle shop. Since then, many successful aircraft

have been designed in an increasingly more complex process, involving both experimental

and analytical or computational models. Testing a new design, or a component of it, in

wing tunnels or by using mathematical models is necessary because of the high risk and

considerable costs involved in producing and flight testing a prototype. With more and more

complex aircraft, longer development cycles and immense development costs, the need for

reliable, accurate, and practical modelling increases.

Wind tunnel testing provides realistic measurements to validate and verify the accuracy of

simulations, and to investigate critical effects that are not capture in simulations. However,

wind tunnel tests suffer from issues such as blockage, scaling, mount interferences and


Mach/Reynolds number effects, ’Beyers (1995)’. The disadvantages of wind tunnel testing

are that:

• Designing and manufacturing a fully equipped wind tunnel model and running the

experimental campaign is very expensive and time consuming. The costs and time

scales involved in testing do not meet the requirements of conceptual and preliminary

aircraft design, where the speed of the investigations and the flexibility in new aircraft

designs are needed.

• For wind tunnel testing, an aircraft configuration is required. If the geometry is not

finalised, testing is of no much value. Wind tunnel testing remains too expensive and

time consuming to provide useful indications to the aircraft designer, and it is usually

performed in the later stages of the aircraft development process when the flexibility

in the design changes is limited.

• Wind tunnel testing is limited by physical constraints, affecting the range of flight

conditions to be tested. Dynamic tests, in particular, are limited by the weight and size

of the wind tunnel model.

Figure 1.8 shows the wind tunnel models of the SACCON and TCR models mounted in

the test facilities. The scaled models were tested at low and high speeds for both static and

dynamic cases.

(a) SACCON model (b) TCR model

Figure 1.8 Wind tunnel testing for the SACCON, ’Vallespin et al. (2011)’, and TCR mod-

els, ’Rizzi et al. (2011)’

Flight Testing

Flight testing is the process of developing and gathering data during operation and flight of an

aircraft and then analysing that data to evaluate the flight characteristics of the aircraft. Flight

test assessments take place in areas which include, but may not be limited to, the following:


• Aircraft Performance. Aircraft performance comprises stall speed measurement, climb

rates and gradients, cruise range and endurance, descent or glide rates and gradients,

and take–off and landing distances.

• Flight Handling. Aircraft flight handling qualities include stability, controllability and

manoeuvrability, trim, and stalling and spinning characteristics.

• Aircraft Systems. Piloting and operational features of aircraft controls, systems and

avionics, e.g. the human–machine interface. Assessment of aircraft instrumentation

and independent test instrumentation used during flight testing is also included.

• Human Factors. Ergonomic, workload and operational environment aspects.

Flight testing is certainly the most accurate and expensive method to obtain information

of a test aircraft. Not only an aircraft prototype is needed but also high risk is involved with

testing, see Figure 1.9. Flight testing has been traditionally used to improve the aerodynamic

predictions obtained through other methods. Parameter identification techniques have been

used extensively to obtain aerodynamic coefficients for aircraft. The general idea behind

parameter identification is to minimize a pre–defined error between the measured and

modelled aerodynamics. Corrections obtained through this error minimization process are

then applied to the modelled aerodynamics to improve the match with the flight test measured

parameters.

Figure 1.9 The MQ–4C Triton test aircraft makes its approach for landing March 13 at Palmdale,

California, marking the conclusion of initial flight testing (Photo by Alan Radecki)

1.3 Generation of Tabular Aerodynamic Model

Refer to Table 1.1 and consider an aircraft with three traditional control surfaces (elevator,

aileron, and rudder). Then, assume a coarse approximation of the flight envelope that consists

of ten samples uniformly distributed for each parameter range. The parameter space, in this

case, spans nine dimensions, e.g. three flight conditions, three control settings, and three

angular rates.

Two situations may arise. The first situation is when there is the requirement to deliver

a table detailing the complete dependence of the aerodynamic coefficients on the nine


parameters. In this case, the tabular aerodynamic model consists of 109 entries, which

corresponds to a matrix of 10 billion rows and 15 columns (nine parameters and six

aerodynamic coefficients). The second situation is for a requirement of a less coupled

representation of the aerodynamic coefficients, as discussed in Section 1.2.1. In this case,

using Equation (1.4), the size of the tabular aerodynamic model is 7,000 entries (reduced

from 10 billion in the first situation).

This Section discusses efficient methods to generate high–dimensional large tabular

models, as exemplified by the problem above.

1.3.1 Brute–force Approach

The costs of computing every single entry of the tabular model are prohibitively expensive.

For the problem above, assume that a single prediction of a rapid numerical model takes

about 10 seconds, inclusive of pre– and post–processing times. When used for to the smaller

case of 7,000 entries, the generation of the full tabular model would require nearly 20 hours

of computing time – an unreasonable time.

When using the brute force approach in combination to high–fidelity aerodynamic models

to fill the tables, an unrealistic time of 158 years was suggested ’Rogers et al. (2003)’. An

alternative to the brute force approach is based on sampling, reconstruction and data fusion

of aerodynamic data, as fully discussed in the following Section.

1.3.2 Surrogate Model

Kriging Model

Kriging method generates an interpolation model for nonlinear and multi-–dimensional

deterministic functions. In ’Cristofaro et al. (2014); Mackman et al. (2013); Zhang et al.

(2013)’ the Kriging interpolation is efficiently used to reduce the computational cost for

generating a full aerodynamic model. Once the Kriging model is created, it becomes a

computationally cheap model for prediction of the function at untried locations.

Consider the three dimensional table for an aerodynamic force/moment coefficient with

the independent variables being α, M and β. To begin, the range of the independent

variables for use in the aerodynamic data table is defined and initial data sample locations

are selected to lie along the boundaries of the three dimensional parameter space. Latin

Hypercube Sampling ’Giunta et al. (2003)’, a modification of the well known Monte Carlo

method ’Hastings (1970)’, is then used to obtain a few additional data sample locations within

the parameter space. The aerodynamic force/moment coefficient is then computed for these

data locations. The idea is that this initial sampling will provide a quick overview of the

variation of the aerodynamic/force moment coefficient throughout the parameter space and

help, as follows, to identify additional data sample locations.

Kriging is then used to interpolate data at untried locations, i.e. where data is not available,

throughout the parameter space. Based on predefined criteria, a new data sample location is

selected in one of the untried points and the process then repeats until a predefined tolerance

on the criteria is met. Selection of the initial data sample locations at the boundaries removes

the risks of extrapolation.


Kriging predicts the values at untried locations using a weighted average of the values

at the available data sample locations. It is different from the other weighted average

interpolation methods in that it assumes that the parameter being interpolated for is a random

variable, it ensures that the expected value of the prediction error is 0 and it minimizes the

variance of the prediction error. Let X = [X1, X2, ..., Xn] be a vector representing data at

known data sample locations s = [s1, s2, ..., sn]. Kriging assumes that the random variable

can be written as:

X(si) = f(si) + Z(si) (1.8)

where f(si) is usually called the mean and represents a trend (constant, linear or quadratic)

and Z(si) represents a stochastic process with variance σ2.

Various modifications of Kriging interpolation method are available in the litera-

ture ’Cressie (1991)’. ’Cristofaro (2014); Ghoreyshi et al. (2009)’ employed Universal Krig-

ing which assumes f(si) is non–constant and a function of the data sample locations. For

Universal Kriging, where each data sample location can be characterized by p parameters:

f(si) =

p∑

k=0

µkfk(si) (1.9)

f(si) has to be defined to solve the Universal Kriging equations. Ideally, this would be

dictated by the physics of the problem. Often, the trend is unknown and is usually modelled

as a lower order polynomial of the coordinates of the design sites. The Kriging prediction, at

an unknown location su, can be written as:

X(su) =

n∑

i=1

wiX(si) (1.10)

where X(su) is the predicted value and wi are the weights to be computed. Note that wi are

allowed to vary for different predictions.

The prediction error at the unknown location, su, can be written as:

R(su) = X(su)−X(su) (1.11)

where X(su) is the prediction and X(su) is the random variable modelling the true value.

Note that the prediction error is also a random variable since it is a linear combination of

random variables.

For the expected value of R(su), ER(su), to be 0:

ER(su) = EX(su)−X(su)

= E

n∑

i=1

wiX(si)

− EX(su)

=

n∑

i=1

wiEX(si) − EX(su)

(1.12)


Substituting for EX(si) and EX(su), we can re-write Eq. (1.12) as:

n∑

i=1

wiEX(si) − EX(su) =

n∑

i=1

wif(si)− f(su) (1.13)

Let us assume that the mean, f(si), is a linear function of α, M and β. This allows us to

write:

f(si) = µ0 + µ1α(si) + µ2M(si) + µ3β(si)

f(su) = µ0 + µ1α(su) + µ2M(su) + µ3β(su)(1.14)

Substituting Eq. (1.14) into Eq. (1.13) we find that, for ER(su) to be 0, we require:

n∑

i=1

wi = 1

n∑

i=1

wiα(si) = α(su)

n∑

i=1

wiM(si) = M(su)

n∑

i=1

wiβ(si) = β(su)

(1.15)

which basically requires∑n

i=1 wi = 1.

As mentioned earlier, Kriging also attempts to minimize the variance of the prediction

error. The variance of a weighted linear combination of random variables can be written as

(’Isaaks and Srivastava (1989)’):

Var

n∑

i=1

wiX(si)

=

n∑

i=1

n∑

j=1

wiwj CovX(si)X(sj) (1.16)

Applying Eq. (1.16) to Eq. (1.11), we can write:

VarR(su) =CovX(su)X(su) − CovX(si)X(sj)

− CovX(su)X(su)+CovX(su)X(su)(1.17)

The first term in Eq. (1.17) describes the covariance of X(su) with itself and is equal to

the variance of X(su). Since X(su) is a linear combination of random variables, applying

Eq. (1.16) to Var∑n

i=1 wiX(si) yields:

VarX(su) = Varn∑

i=1

wiX(si) =n∑

i=1

n∑

j=1

wiwjCij (1.18)


Similarly, the last term in Eq. (1.17) describes the covariance of X(su) with itself and is

equal to the variance of X(su). As mentioned earlier, that the random variables modelled by

Kriging are assumed to have variance σ2. Hence:

VarX(su) = σ2 (1.19)

The remaining two terms in Eq. (1.17) can be lumped together as 2CovX(su)X(si)and simplified as follows:

2CovX(su)X(su) = 2Cov

((

n∑

i=1

wiX(si)

)

X(su)

)

= 2

[

E

(

n∑

i=1

wiX(si)X(su)

)

− E

(

n∑

i=1

wiX(si)

)

E(X(su))

]

= 2

[

n∑

i=1

wiE(X(si)X(su))−

n∑

i=1

wiE(X(si))E(X(su))

]

= 2n∑

i=1

wi[E(X(si)X(su))− E(X(si))E(X(su))]

= 2

n∑

i=1

wiCiu

(1.20)

Note that the derivation of this equation makes use of the theorem ’Isaaks and Srivastava

(1989)’ that CovX(su)X(su) = EX(su)X(su) − EX(su)EX(su).

Combining the expressions from Eq.(1.18), Eq.(1.19) and Eq.(1.20), we can write the

expression for the variance of the prediction error as:

σ2R =

n∑

i=1

n∑

j=1

wiwjCij − 2

n∑

i=1

wiCiu + σ2 (1.21)

The minimization of Eq. (1.21) is a constrained problem with constraints given in

Eq. (1.15). To find a solution, the method of Lagrange multipliers is used to transform this

constrained problem into a unconstrained one ’Isaaks and Srivastava (1989)’:

σ2R =

n∑

i=1

n∑

j=1

wiwjCij − 2

n∑

i=1

wiCiu + σ2 + 2

p∑

k=0

µk

(

n∑

i=1

wifk(si)− fk(s0)

)

(1.22)

The error variance presented by Eq. (1.22) can be minimized by setting the first partial

derivatives of n+ p+ 1 equations with respect to wi and µk to 0. This results in the

n+ p+ 1 equations which can be written in the form Ax = B as follows:

[

Cij fik

fjk 0

] [

wj

µk

]

=

[

Ci0

f0k

]

(1.23)


where:

Cij =

C11 . . . C1n

.... . .

...

Cn1 . . . Cnn

, fi

k =

1 f11 . . . fn

1

......

. . ....

1 f1p . . . fn

p

, fj

k = [fik]′

wj =

w1

...

wn

, µk =

µ0

...

µp

, Ci0 =

C10

...

Cn0

, f0

k =

1

f01

...

f0p

To solve the universal kriging equations, a covariance model for the data is

required. Several covariance models are available in the literature, see for example

’Isaaks and Srivastava (1989)’.

Example 1

This example demonstrates a simple use of the Kriging model. It shows how an interpolation

model can be obtained and how it can be used to compute the approximated value of the

function in a chosen point. Let us start by considering a one–dimensional functionX = f(s),whose value are known only in n specific points. Target of the Kriging model is the evaluation

of the function in unsampled locations.

Given a set of n = 3 design sites, s = [s1, . . . , sm] = [−2,−1, 3], and responses X =[X1, X2, X3], we aim to compute the value X(su) in su = 0. A schematic representation of

the problem is given in Figure 1.10.

−2 −1 0 3

X1 X2 X(su) X3

Figure 1.10 Problem domain, the known data are X1, X2 and X3 and the unknown is X(su).

A covariance, or correlation, model is needed to describe the relationships among

variables. So let consider a linear model defined as R(d) = max0, 1− θ · d with d the

relative distance. We take the parameter θ = 1/6 = 0.167, so that all the considered points

influence each other. Since the smaller is θ, the wider is the influence of known data over the

domain, this value need to be chosen considering the nature of the problem. The resulting

values of the covariance model are then presented in Table 1.2.

Table 1.2 Values of a linear covariance model with θ = 1/6.

d 0 1 2 3 4 5 6 . . .

R(d) 1 0.833 0.667 0.5 0.333 0.167 0 0


The matrix with the covariance model values between the sampled locations, C, is then

needed. The elements indicate the respective correlation between different sampled points

Cij = R(|xi − xj |). Equation (1.24) shows how the correlation matrix is created within this

example. The C matrix is squared with dimension n× n and it is dependent on the norm of

the distance between samples points, explaining the symmetrical nature of the matrix.

C =

1 0.833 0.1670.833 1 0.3330.167 0.333 1

(1.24)

As done with C, a covariance vector, D(su), between sampled points and a generic

unknown point su, must be generated as showed in Eq. (1.25). The elements indicate the

influence of known data on the unknown point Di = R(|si − su|). Considering su = 0, the

first element is D1 = R(|s1 − su|) = R(2) = 0.667.

D(su) =

R(|s1 − su|)R(|s2 − su|)R(|s3 − su|)

−→ D(0) =

0.6670.8330.5

(1.25)

It is now possible to find the weights of the available samples on the unknown points, w. The

theory gives the resulting formula as presented in Eq. (1.26), where 1 is a ones vertical vector

of size n.

It is important to notice that since C is not function of su, the inversion of the matrix is

necessary only once, even when many estimations are required.

w(su) = C−1

D(su)−C−1 ·

(

D(su)TC

−11− 1

)

·(

1TC

−11)−1

· 1 (1.26)

The resulting weights are then presented in Figure 1.11.

−2 −1 0 3

0 0.75 ⋄ 0.25

Figure 1.11 Resulting Kriging model weights in s1, s2 and s3 associated with su = 0.

The resulting value in su are then easily computed as X(su) = w(su)′ ·X.

Now let consider two data cases: one linear with X = [0, 1, 5] and one parabolic with

X = [0, 1, 25]. The resulting value X(su) in su = 0 are respectively 2 for linear and 7 for

parabolic, as showed in Figure 1.12.

The reader is finally invited to reflect on the following points:

1. Since we used a linear correlation model, for the linear case the predicted value is

actually coincident with the real value.

2. About the parabolic case the predicted value is a linear interpolation between the values

in 0 and 3. In this case the Kriging method neglects the point in -2, obtaining there a

weight of 0 for su = 0.


s

X(s)

−2 −1 0 1 2 3

1

2

3

4

5

6

7

X1

X2

X3 linear

•

X0 linear

•

X0 parabolic

ǫ

Figure 1.12 Example of the Kriging interpolation, results for a parabolic and a linear function with a

linear covariance model.

Example 2

In this example, we illustrate the use of Kriging to iteratively refine the sample space for the

lift coefficient of NACA 0012 airfoil in the α−M domain. Let consider the angle of attack

ranging between 0 deg and 14 deg and the Mach number between 0.3 and 0.7.

Figure 1.13 presents a schematic of the code UniversalKriging that is ancillary material

with this chapter. To begin with a total of six sample points are selected, four of which

lie at the boundaries and two of which are selected to lie in the regions where we expect

nonlinearities. The points at the domain boundaries allow to avoid extrapolation. The two

other points can be selected via a monte carlo simulation. However, if a priori knowledge of

the physical phenomenon governing the sample space is available, these points can also be

selected through engineering judgement, as is in this case. Table 1.3 presents the details for

the initial samples.

The goal is to iteratively select more samples with the objective of capturing the

nonlinearities while keeping the number of samples to a predefined limit. The choice about

the new sample location is driven by the maxMSE method.

In the first step the initial samples are selected and computed. The selection of a

covariance model and a regression model are then necessary in order to compute a Kriging

model. In this example an exponential covariance model is used (i.e. Cov = C0(e−θ|h|

a ),


Table 1.3 Initial samples for this example

Sample α M Cl

1 0 0.3 0

2 14 0.3 1.3626

3 9 0.5 0.9318

4 8 0.6 0.7822

5 0 0.7 0

6 14 0.7 0.6326

’Isaaks and Srivastava (1989)’), and a linear regression model, presented in Equation 1.14.

The Kriging ROM model computation can start. The samples and independent variables

are first normalized and the covariance of the samples with respect to each other and with

respect to the design sites is computed with the chosen covariance model. The matrices

Cij , Ci0, fik and f0k are assembled, as showed in Equation 1.23. The Kriging weights

wj and Lagrange multipliers µk are obtained from the solution of the linear system in

Equation 1.23. Finally the error variance, σ2 is computed in every domain point with the

expression presented in Equation 1.22.

The maxMSE sampling method chooses as next sample the location where the error

variance is maximum. The independent parameters on the new sample location are then

computed and the values added to samples set.

The Kriging ROM model is then recomputed and the process goes on until a stop criterion

is obtained.

1.3.3 Adaptive Design of Experiment

Off–the–shelf Algorithm

Statistical criteria are particularly popular owing to the widespread use of Kriging models.

Maximizing the entropy, or amount of information provided by a sample, is the simplest

and most intuitive of these methods, involving successively adding sample points at the

locations with the largest value of error predicted by the Kriging mean squared error (MSE)

function, ’Shewry and Wynn (1987)’. This will be referred to as the entropy or MaxMSE

criterion. MaxMSE samples are primarily space–filling but adapt to the relative variability of

each coordinate direction as well, because the MSE function is dependent on both the sample

positions and the Kriging model parameters.

This method can be used to easily create a sampling procedure and it has been successfully

employed for the aerodynamic table generation in ’Cristofaro et al. (2014)’. As benchmark

for this method, the aerodynamic data from ’Vallespin et al. (2011)’ for the UCAV and

from ’Rizzi et al. (2011)’ for the TCR configuration are used. These data come from CFD

solutions, obtained on a equally spaced domain. Figure 1.14 shows the UCAV lift and

pitching moment coefficients predicted via MaxMSE criterion. The presented data are

obtained after 10 iterations (plus 2 initial samples at the extrema).

The mean relative error of the pitching moment prediction compared to the target function

is 3.47%, however it does not capture the strong function nonlinearity.


Initial sampling

Select a covariance model

Select a model for the sample mean

Normalize the samples and

the independent parameters

Compute covariance

Set up matrices

Solve the linear system

Compute the error variance

New design site selection

and computation

Figure 1.13 Flow chart of the implemented code UniversalKriging.

In Figure 1.15 the MaxMSE method for the pitching moment coefficient of the TCR

configuration is presented. The colored surface indicates the Kriging model prediction

obtained with the sample points, red spheres. The points show the discrete target function.

The mean relative error of the pitching moment prediction compared to the target function is

15.90%.

The algorithm provided by the author called UniversalKriging shows the full procedure

for the Kriging model generation and the MaxMSE application. In Figure 1.15 the samples

tend to a uniform distribution. During the sampling process this is evident: for the one–

dimensional case, the method first chooses the middle point, then the two at 1/4 and 3/4 and

so on. Since 10 iterations were run, all the points are located at multiples of 1/16 along the

domain. However this algorithm does not consider the shape of the function.

An algorithm that considers the output function shape and so the point potential importance

over the model accuracy, is the expected improvement function. The EIF is a statistical

criterion, developed for efficient global optimization by ’Jones et al. (1998)’. It leads to


(a) Lift with MaxMSE criterion (b) Pitching moment with MaxMSE criterion

Figure 1.14 MaxMSE criterion applied to lift and pitching moment coefficient of the UCAV testcase

Figure 1.15 Pitching moment with MaxMSE criterion of the TCR after 10 iterations (with 4 initial

samples at the domain vertices)

points that maximize the expectation of improvement upon the global minimum or maximum

of the current predictor. It is generally used in combination with the MaxMSE criterion,

however it only consider global maxima and minima, neglecting all the local ones. In

Figure 1.14 is evident that the strong nonlinearity is close to local maxima and minima, so

that concentrating the resources to global maximum would not lead to the most efficient

result.


1.3.4 Cognitive Sampling Algorithm

A new approach was introduced by ’Cristofaro (2014)’ with different sampling criterion.

Two algorithms were developed considering the aerodynamic table generation problem.

The computational budget is in the order of tens and the target of the criteria is to focus

the resources on the unpredictable nonlinear part of the functions, e.g. stall or bow shock

appearing, partially neglecting the linear part. The aerodynamic nonlinearities exhibit in the

aerodynamic forces trends with slope changing, and so local maxima or minima and high

curvature values are distinctive of nonlinearities appearing.

The toolbox CognitiveSampling, ancillary material with this chapter, encloses the

developed algorithms and various test cases examples.

Local maxima and minima criterion

The criterion based on local maxima and minima is a generalization of the EIF criterion for

which only global maxima and minima are searched. The new method finds the position

of all local maxima and minima of a Kriging surrogate full model generated with the

available samples. These domain locations are computed comparing any function value

with all the points inside a sphere centered in it (left and right points for one dimension

domain problems). The sphere radius is initially computed as minimum of the Euclidean

norms of any two points with all non–equal coordinates. If the function value is bigger

or smaller than all the others in the sphere, the point is marked as local maximum or

minimum, respectively. After that all the local maxima and minima of the surrogate model

are found, the prediction error of the Kriging model, mean squared error mse, is used. For

every maximum and minimum, the point with maximum mse value, belonging to a sphere

centered in the considered location and with radius equal to the distance from the nearest

sample point, is extracted. Then the one with the maximum mse value is extracted between

them. Finally the mse of the found point is compared with the global mse maximum divided

by a custom scale and so the algorithm decides where to locate the next sample as the

biggest between them. The choice of looking only in sphere of radius equal to the maximum

distance from the closest sample is not optimal, because in other directions the error may still

increase. However if the samples budget is small, the algorithm may not look over the whole

domain, neglecting the most important nonlinearities. This issue is partly avoided by always

comparing the found point and the global maximum error scaled by a custom factor. In this

way the computation does not blindly persist on the same location.

Second derivative criterion

An important numerical value indicating the difficulty in predicting the real function with

an interpolation model is the function curvature. The fastest the function changes, the

highest the curvature is and the most difficult is obtaining a good prediction. The second

criterion included in the toolbox is called Hessian and is based on this consideration.

Starting from the available function samples, an interpolating Kriging model is generated.

The approximated full model allows to compute a second derivative approximation with a


central difference (second order of accuracy) as shown in Equation (1.27).(

∂2y

∂x2

)

i

≃yi−1 − 2 · yi + yi+1

∆x2(1.27)

The most important location is then evaluated as the point with the maximum mse

belonging to a sphere centered in the point with the highest curvature and with radius equal

to the distance from the nearest already sampled point. Finally the mse of the found point is

compared with the global maximum of the mse divided by a custom scale and so the algorithm

decides where to locate the next sample as the biggest between them. The Achilles’ heel of

the second order derivative based criterion are the points with non–continuous first derivative

(cusps), in which the central difference returns very high values.

For n–dimension domain problems, a unique second derivative value is not available, but

it has different values when computed along different directions. In order to consider all the

local second derivative values and not to be bounded to any frame of reference, an easy but

reasonable way is then found. The global problem is to find where a generic discrete function

f exhibits the biggest curvature in space, with respect to any considerable direction. The

Hessian matrix elements are defined as partial derivatives as shown in Equation (1.28a). If

the second derivatives of f are all continuous, then the Hessian is a symmetric matrix (for

the symmetry property of second derivatives known as Schwarz’s or Clairaut’s theorem).

H =

∂2f∂x1∂x1

∂2f∂x1∂x2

. . . ∂2f∂x1∂xn

∂2f∂x2∂x1

∂2f∂x2∂x2

. . . ∂2f∂x2∂xn

......

. . ....

∂2f∂xn∂x1

∂2f∂xn∂x2

. . . ∂2f∂xn∂xn

(1.28a)

Hi,j =∂2f

∂xi∂xj

=∂

∂xi

(

∂f

∂xj

)

(1.28b)

A finite difference approximation of the Hessian matrix is adopted. About the

diagonal terms, pure second derivative, the approximation with central difference presented

in Equation (1.27) is used, obtaining a second order of accuracy. The procedure of

approximating the mixed second derivatives is based on central difference (second order of

accuracy as well) and it is fully illustrated in Eq. (1.29a). The i and j stand for the indices of

the discrete domain indicating the xi and xj coordinates of the analyzed point.

(

∂2f

∂xi∂xj

)

i,j

=

(

∂

∂xi

(

∂f

∂xj

))

i,j

≃

(

∂

∂xi

(

fj+1 − fj−1

|xj+1 − xj−1|

))

i

≃

(

fj+1−fj−1

|xj+1−xj−1

|

)

i+1−(

fj+1−fj−1

|xj+1−xj−1

|

)

i−1

|xi+1 − xi−1|(1.29a)


The Frobenius norm of the Hessian matrix, presented in Equation (1.30), can then be used

as numerical index of the function curvature. This value is not dependent on the used frame

of reference, because the Frobenius norm is invariant under a unitary transformation, as a

reference frame rotation.

‖H‖F =

√

√

√

√

n∑

i=1

n∑

j=1

H2ij (1.30)

In case of a multi outputs function a criterion was developed in order to give priority to

some outputs about the choice of the next sample location. In case a very fast and efficient

computation is needed, the user may prefer to focus on the main aerodynamic loads as lift,

drag or pitching moment and decide to have a more precise result about them, neglecting the

others.

Test cases applications

Two real cases are now considered in order to obtain a validation on real data sets. The

SACCON and the TCR test cases are presented in Section 1.1.5.

In Figure 1.17 some aerodynamic coefficients of the UCAV SACCON are presented. The

lift, drag and pitching moment coefficients with respect to the angle of attack are considered.

The analysis conducted is one–dimensional.

Both the sampling criteria available in the toolbox are used. The solutions give very good

results, finding the sudden drop of the pitching moment with only 12 sample points for both

methods (iteration number 10). About the maxima–minima criterion this is achieved because

of the presence of a local maximum close to the nonlinearity in the pitching moment. About

the second derivative based criterion the pitching moment drop is investigated thanks to an

high second derivative in the lift function at the same location, permitting to have a further

analyses close to this nonlinearity. In these cases the mean relative error of the pitching

moment prediction compared to the target function is respectively 2.40% with the maxima–

minima criterion and 1.92% with the second derivative criterion. Both the prediction models

predict the sudden and narrow aerodynamic nonlinearity manifesting with a pitching moment

fall.

The sampling methods are then applied to the TCR test case. In Figure 1.17, the colored

surface indicates the Kriging model prediction obtained with the sample points, red spheres.

The diamonds shows the discrete target function. The mean relative error of the pitching

moment prediction compared to the target function is respectively 15.90% for maxima–

minima criterion, and 13.12% for Hessian criterion. It can be seen how the two methods

predict well the general trend of the function with only 14 samples (4 starting and 10

iterations). The relative error is relatively big, however only 14 points over 318 total points

of the domain were analyzed (4.4% of the total required computations).

1.3.5 Data Fusion

The aerodynamic coefficients can be generally obtained using different sources. If more data

sets are available, data fusion can combine them in order to obtain a more accurate model.

In the case of two data sets available, usually one can be considered low–fidelity (lf, cheaper

and usually more populated) and the other high–fidelity (hf, more expensive and usually less


(a) Lift with local maxima–minima criterion (b) Pitching moment with local maxima–minima crite-

rion

(c) Lift with second derivative criterion (d) Pitching moment with second derivative criterion

Figure 1.16 Comparing the results obtained with the two developed methods on the UCAV data after

10 iterations (2 initial samples at the extrema), ’Cristofaro (2014)’

populated). During the data fusion process the cheap samples provide information about the

trend of the target function, while the expensive samples give quantitative information of the

model. Two methods are considered to apply data fusion:

1. One is explained in ’Da Ronch et al. (2011a)’. A Kriging interpolation model η(x) is

calculated from the samples of the cheap aerodynamic evaluations and it is evaluated

at the locations at which expensive predictions are available, η(xi). The vector of the

input parameters at the expensive samples, xi, is then augmented by the evaluation of

the Kriging function for the cheap samples: xaugi = [xi η(xi)]. A Kriging interpolation

model is finally calculated for the augmented samples and the data fused evaluation is

given by the evaluation of such function in the continuous vector xaug = [x η(x)].


(a) Pitching moment with local maxima–minima crite-

rion

(b) Pitching moment with Hessian criterion

Figure 1.17 Comparing the results obtained with the two developed methods on the TCR data after

10 iterations (with 4 initial samples at the domain vertices), ’Cristofaro (2014)’

2. Another method is described in ’Santini (2009)’. As the previous algorithm a Kriging

interpolation model η(x) is calculated from the samples of the cheap aerodynamic

evaluations. This criterion is then based on an increment function β(x) obtained

with the interpolation of β(xi) = fhf(xi) − η(xi) where fhf(xi) are the high–fidelity

sampled points. From this the data fusion approximation is easily derived as

f(x) ≃ η(x) + β(x).

The first method presents some oscillating problems dealing with the interpolation for a

second order regression model because of the aligned nature of xaugi and in the case of

a linear low–fidelity database, the resulting Kriging interpolation matrix is ill–conditioned.

For these reasons the implemented function try to use this algorithm, but it does not always

succeed. So the user is advised to use the second method and be very careful about the first.

Two similar data fusion cases are investigated using the second of the methods previously

described. The high–fidelity data are taken from results of the wind tunnel test data about a

UCAV reported by ’Vallespin et al. (2011)’ already used in the one–dimensional validation.

The sample point choice is taken from the results of the previously described Hessian iterative

sampling criterion. About the low–fidelity data, in the first case a custom function presenting

an initially linear behaviour and then a sudden drop, is used. This function is slightly

vertically shifted with respect to the high–fidelity data. The second low–fidelity case is

taken from ’Vallespin et al. (2011)’ and represents the resulting data of a CFD computations

solving RANS equations with a k–ǫ turbulence model. This case is more realistic and

represents a real possible situation.

The first result presented in Figure 1.18 shows that, in case of a linear low–fidelity data

trend, the data fusion looks very similar to a linear interpolation of the high–fidelity data.

A small difference may appear close to the high curvature of the low–fidelity function, for

which the fused data try to represent the same bend. The second resulting fused data exhibits


(a) Custom low fidelity function (b) CFD obtained low fidelity function

Figure 1.18 Data fusion applied to the UCAV wind tunnel high fidelity data with low two different

fidelity data, ’Cristofaro (2014)’

a strong nonlinearity. The two data sets may appear similar but, because of the horizontal

shifting of the curves, the fused data displays a high oscillation close to the presence of

the nonlinearity. Theoretically we do not know which is the exact position of that, but the

low–fidelity data information may lead astray. In conclusion data fusion may bring some

improvement, but the user is highly advised to take a critic look at the results.

Data fusion can be inserted during the iterative sampling algorithm, obtaining a more

accurate model of the function starting from the first iterations. The results after 3 iterations

using data fusion with the two previously described low–fidelity databases and not using

data fusion are presented in Figure 1.19. Figure 1.19 shows that the prediction follows the

low–fidelity database. About the first case the ease of the cheap available results is that the

resulting fused data are well approximated, whereas the nonlinearity is not found. The second

case instead samples the nonlinear part, because of its presence in the cheap available results,

but the overall approximated function is a worse approximation. For this reason the user is

advised to use data fusion during iterative sampling, but only for the first iterations, otherwise

the low–fidelity database might bring to some inaccurate prediction.

1.3.6 Prediction of Dynamic Derivatives

Dynamic derivatives are calculated from observing the response of aerodynamic forces

and moments to translational and rotational motions. Dynamic derivatives influence the

aerodynamic damping of aircraft motions and are used to evaluate the aircraft response and

the open–loop stability, e.g., short–period, Phugoid and Dutch roll modes.

A common wind–tunnel testing technique for the prediction of dynamic derivatives relies

on harmonic forced–oscillation tests. After the decay of initial transients, the nature of the

aerodynamic loads becomes periodic. A time–domain simulation of this problem requires

significant computational effort. Several oscillatory cycles have to be simulated to obtain a


(a) Data fusion with custom low fidelity function (b) Data fusion with CFD obtained low fidelity function

(c) No iterative data fusion

Figure 1.19 Iterative data fusion applied to the UCAV wind tunnel high–fidelity data after 3 iterations

different low–fidelity databases, ’Cristofaro (2014)’

harmonic aerodynamic response, and a time–accurate solution requires a small time–step

increment to accurately capture the flow dynamics ’Da Ronch et al. (2012); Mialon et al.

(2011)’. Time–domain calculations support a continuum of frequencies up to the frequency

limits given by the temporal and spatial resolution, but dynamic derivatives are computed at

the frequency of the applied sinusoidal motion.

It is therefore worthwhile to consider a frequency–domain formulation to obtain a

good estimate of the derivatives at reduced computational cost. These computational

approaches, based on the Harmonic Balance (HB) and the Linear Frequency Domain (LFD)

methods, provide the ability to efficiently approximate the aerodynamics resulting from

small, periodic and unsteady perturbations of the geometry of an aircraft configuration. By

resolving only the frequencies of interest, the computational cost of dynamic derivatives is


greatly reduced. Initially developed in the field of turbomachinery ’Clark and Hall (2000);

van der Weide et al. (2005)’, the HB and LFD methods have been subsequently used for

external aerodynamics applied to aircraft problems ’Da Ronch et al. (2010, 2013)’. A large

amount of research has been devoted to applications of the HB and the LFD technologies

to a broad spectrum of engineering disciplines. There is the question of the influence of

the approximations on the derivative predictions. A body of previous work ’Da Ronch et al.

(2013); Mialon et al. (2011)’ looked at the adequacy of frequency domain methods for

the rapid calculation of dynamic derivatives for use in flight dynamics analysis. Thorough

investigations of the dependencies of dynamic derivatives on model parameters were

performed at realistic flight conditions, and the limitations of the tabular aerodynamic models

traditionally used by flight dynamicists were assessed ’McCracken et al. (2012)’.

Method of data analysis

Two techniques to post–process time–domain sampled data obtained from unsteady time–

domain simulations are common. First, the transformation to the frequency domain is

considered to gain insights into the frequency spectra of aerodynamic loads. To improve

over standard techniques, the method described in ’Da Ronch et al. (2012)’ can be adopted.

Second, a regression–based approach can be developed whereby the unknown vector of the

approximate solution is obtained from the minimisation of an appropriate functional. More

details can be found in ’Da Ronch et al. (2013, 2012)’.

Finally, note that an implementation of a high–accurate Finite Fourier Transform (FFT) is

provided with this Chapter. A comparison between the novel approach and the standard FFT

is performed for two test cases.

1.4 Time Accurate CFD for Flight Simulation

CFD is becoming credible for the computation of aerodynamic time history effects. The

flight dynamics of a manoeuvering aircraft could potentially be modeled by coupling

the unsteady Reynolds–averaged Navier–Stokes (URANS) equations and the dynamic

equations governing the aircraft motion. First attempts were limited to two–dimensional

test cases, while recently the coupled CFD–flight dynamics of a full aircraft has been

studied ’Ghoreyshi et al. (2011a)’. However, the CFD–flight dynamics simulations take

substantially longer than when taking the forces and moments from look–up tables raising

the open question of when time history of the aerodynamics is required for flight dynamics

analysis. A framework for investigating the limits, from flow unsteadiness, of aerodynamic

tables is described in ’Ghoreyshi et al. (2011a)’. The particular demonstrations of this

framework are made for a generic (and sharp leading–edge) fighter performing time–optimal

maneuvers with the aerodynamics given by the Euler equations.

The key functionality for the CFD solver is the ability to move the mesh. Two types of

mesh movement are required. First, a rigid rotation and translation is needed to follow the

motion of the aircraft. Secondly, the control surfaces are deflecting throughout the motion,

which can be achieved either by mesh deformation or a Chimera technique ’Madrane et al.

(2004)’.

’Ghoreyshi et al. (2011a)’ shows the comparison between the prediction of the

aerodynamic forces from the static and dynamic tabular model and the motion replay in the


CFD framework. This is done for slow motions where close agreement would be expected.

The motions used are trimmed level flight, pull–ups with constant and varying angle of attack,

wingover and 90 deg turns. The comparisons test the CFD formulation of the maneuver

replay, which is done in a time accurate fashion with control surface deflections.

A number of slow longitudinal and lateral maneuvers were used to demonstrate that the

tabular and replay aerodynamics agreed closely as expected. The cost of generating the

static tables was the equivalent of 200 steady–state calculations, and the simulation of each

maneuver was possible in roughly one tenth of this for the cases shown. Then a number of

higher rate maneuvers for a pull–up was used to investigate the influence of dynamic and

unsteady terms. At moderate rates the addition of dynamic tabular terms brought the tabular

predictions into agreement with the replay values. However, for a high angle of attack, high

rate motion, the dynamic terms were not adequate to achieve agreement. A quasi–steady

calculation confirmed that the tabular prediction is correct, and so the disagreement is due

to the influence of the history in the unsteady replay maneuver. This example shows the

usefulness of the framework in investigating the limits of the tabular aerodynamic models.

Some results comparisons for slow, medium and fast rate pull–ups manoeuvres are hereby

presented in Figure 1.20. The resulting lift coefficients are shown together with the angle of

attack (AoA) time history. In the slow pull–up manoeuvre, the angle of attack increases from

0 deg to 12 deg in 10 seconds. Since dynamic effects are expected to be small at these rates,

tabular and replay values of the lift coefficient show a good agreement. About the medium

rate manoeuvre, at low angles the static tabular predictions are close to the replay values,

with significant differences at the higher angles when vortical flow is present. The addition

of the dynamic tabular terms produces a good agreement with the replay values at the low

angles, and a much closer agreement at the higher angles. Finally in the fast manoeuvre is

evident a larger difference between the replay and static tabular predictions at all angles. The

addition of dynamic effects makes the tabular predictions close at the lower angles of attack,

but cannot correct the discrepancy at the high angles when significant history effects due to

vortical interactions are present. Note also the spikes from the replay solution which are due

to dynamic effects from the very rapid motion of the control surfaces.

Based on previous work ’Da Ronch et al. (2012); Ghoreyshi et al. (2011a,b, 2010);

Vallespin et al. (2012)’, tabular models are found inadequate to predict motions at critical

conditions. Attempts to improve over the traditional model have been made, the most notable

being the studies presented in ’Da Ronch et al. (2011b, 2014); Ghoreyshi et al. (2013)’.

1.4.1 Advanced Mathematical Models

The prediction of unsteady nonlinear flow conditions is an important part of the aircraft

design. The ability to predict the aerodynamic loads with some realism within the

aerodynamic design cycle can accelerate industrial design processes and avoid costly design

changes. In need of low computing times, industrial analysis procedures often rely on low–

fidelity numerical aerodynamic methods. However, a large body of work on virtual flight

simulation has revealed that the traditional model based on stability derivatives is inadequate

to model unsteady aerodynamic effects for modern manoeuvring aircraft. In response to the

need of improving the fidelity of simulations while keeping the computational costs low,

more sophisticated models of the aerodynamics have been proposed.

Creating a full–order model for stability and control (S&C) analysis is a computationally


(a) AoA time history for a low rate pull–up (b) CL comparison for a low rate pull–up

(c) AoA time history for a medium rate pull–

up

(d) CL comparison for a medium rate pull–up

(e) AoA time history for a fast rate pull–up (f) CL comparison for a fast rate pull–up

Figure 1.20 Aerodynamic responses for pull–up manoeuvres with tabular models and replay

simulation, ’Ghoreyshi et al. (2011a)’.

expensive approach because such a model needs a large number of coupled computations

for different values of motion frequency and amplitude. An alternative approach to creating


the full–order model is to develop a reduced order model (ROM) that seeks to approximate

CFD results by extracting information from a limited number of full–order simulations.

Ideally, the specified ROM can predict aircraft responses over a wide range of amplitude

and frequency within a few seconds without the need of running CFD simulations again.

Recent efforts on the development of ROMs can be classified into two types: time–

domain and frequency–domain approaches. The frequency–domain models are obtained

from matching transfer functions computed from the measured input–output data. Examples

of the frequency–domain ROMs are the indicial response method by ’Ballhaus and Goorjian

(1978)’ and a frequency–domain approach based on proper orthogonal decomposition (POD)

by ’Hall et al. (1999)’. Some examples of time–domain ROMs include the Volterra theory,

radial basis functions (RBFs), and state–space modeling ’Goman and Khrabov (1992)’.

These ROM techniques have been used extensively for flutter prediction, limit cycle

oscillation, and gust–response modeling, but their application to S&C is still new. Also, only

a few studies have been conducted for reduced–order modeling of aircraft configurations,

mostly limited to the subsonic flow regime. Two techniques, one based on the Volterra series

and one on the nonlinear indicial functions, are reviewed in this Section. Denote y the output

quantity which is of interest in the aerodynamic simulation, and x the input parameter that

influences y. For example, y may refer to the aerodynamic pitch moment, and x to the angle

of attack. The reduced order models try to define a simple yet nonlinear relationship that fits

the dependency of y on x.

Volterra Series

Following the developments of the Volterra theory ’Volterra (1930)’, the output of a

continuous–time, casual, time–invariant, fading memory system in response to the input

vector x(t) can be modeled using the pth–order Volterra series in Equation 1.31.

y(t) = Ψ(x(t)) =

p∑

i=1

Hi(x(t)) (1.31)

where H represents the multi–input Volterra operator. For example, the output response of a

multi–input Volterra series up to second order is formulated in Equation 1.32.

y(t) =m∑

j=1

t∫

−∞

Hxj

1 (t− τ)xj(τ) dτ

+

m∑

j1=1

m∑

j2=1

t∫

−∞

t∫

−∞

Hxj1

,xj2

2

× (t− τ1, t− τ2)xj1(τ1)xj2 (τ2) dτ1dτ2 +O(|x|3) (1.32)

Note that the superscripts in Equation 1.32 identify to which inputs the kernel corresponds;

for example, the second–order kernel Hxj1

,xj3

2 correlates the inputs xj1 and xj3 . Note

that the second– and higher–order kernels are symmetric with respect to the arguments

Hxj1

,xj3

2 = Hxj3

,xj1

2 . The accurate determination of the Volterra kernels is critical for the

generation of an efficient, robust, and nonlinear model. Techniques to identify the Volterra

kernels from CFD calculations are discussed at the end of this Section.


Nonlinear Indicial Functions

Start with a linear indicial function defined by the relationship of Equation 1.33.

y(t) =d

dt

t∫

0

A(t− τ)x(τ) dτ

(1.33)

where A represents the unit response, or indicial function, of the system. For a linear system,

H1 is the impulse response function, and H1(t) = dA(t)/dt applies. The indicial response

functions are used as a fundamental approach to represent the unsteady aerodynamic loads.

For nonlinear aerodynamic responses due to motions starting from different Mach numbers,

the dependencies on the angle of attack and Mach number are added to the indicial functions

in Equation 1.34.

Cm(t) =d

dt

t∫

0

Cmα(t− τ, α,M)α(τ) dτ

+d

dt

t∫

0

Cmq(t− τ, α,M)q(τ) dτ

(1.34)

where M denotes the freestream Mach number. The response function due to pitch rate,

i.e., Cmq(α,M), can be estimated using a time–dependent interpolation scheme from the

observed responses. A possible technique is based on the surrogate modelling approach

discussed in Section 1.3.2 of this Chapter.

System Identification and Results

The generation of reduced models is done by post–processing the output time history in

response to a prescribed input time history. The choice of the most appropriate input signals

is somewhat dependent on the specific applications of the reduced model, and a globally

valid approach seems not to exist. As reported in ’Ghoreyshi et al. (2013)’, common input

time histories, denoted as training manoeuvres, are the linear chirp, spiral, and Schroeder

functions. These training manoeuvres are shown in Figure 1.21.

The test case of ’Ghoreyshi et al. (2013)’ is the X–31 aircraft, a super–manoeuvrable

fighter with a canard wing and a double delta wing. The complex flow features around the

X–31 aircraft are shown in Figure 1.22.

The Volterra ROM predictions based on spiral and chirp training maneuvers are compared

with the time–accurate solution data in Figure 1.23. The comparisons show a good agreement

with CFD data for a ROM identified from spiral data, but the ROM identified from chirp data

does not match well, in particular, around the maximum and minimum angles of attack.

The instantaneous frequency in the chirp maneuver varies with time, and hence it might

not have sufficient information to identify the Volterra kernels corresponding to a swept–

amplitude motion at constant frequency. However, the ROM based on chirp data could be

used for predicting aerodynamics responses from pitch oscillations at many other frequencies

(those covered by the simulation of chirp training maneuver), but the ROM based on spiral is

possibly valid for the motions at a fixed reduced frequency.


(a) Linear chirp (b) Spiral (c) Schroeder

Figure 1.21 Training manoeuvres used to generate Volterra and indicial models, ’Ghoreyshi et al.

(2013)’

The full–order model is compared with the predictions from a linear ROM of the target

maneuver in Figure 1.24(a). Linear ROM fails to accurately predict the pitching moment

values at all angles of attack. The functions of Cmαvary largely with angle of attack at

transonic speed range, and thus a linear ROM cannot predict these effects.

A nonlinear ROM was then created with Equation 1.34 and, using a linear interpolation

scheme, the prediction of the target maneuver was evaluated. Figure 1.24(b) shows that the

nonlinear ROM predictions agree very well with full–order simulation values. Note that such

a nonlinear ROM may be used for computing the pitching moment responses from many

other motions with different amplitudes and frequencies.

Cost of Generating the ROM

The objective of ’Ghoreyshi et al. (2013)’ was to generate cost–effective reduced–order

models capable of predicting aerodynamic loads of an aircraft pitching within the

frequency/amplitude/Mach space of interest. This forms the basis for the future studies of

flight dynamics, where forces and moments are given by these models. Each reduced–order

model used requires a different computational cost and is based on various simplifying

assumptions pertaining to the flow physics. For example, the generation of linear indicial

functions is relatively inexpensive, but the model is limited to small–amplitude motions

at a fixed Mach number for which functions were calculated. A first–order linear Volterra

model also has the same limitations. The nonlinear indicial functions include responses to

different angles of attack and could be used for predicting responses to arbitrary motions

at a fixed Mach number, but they have relatively high computational cost compared with

a linear model. A Volterra model with second and higher–order kernels has the nonlinear

dependencies of aerodynamic loads with amplitude. However, the suitability of the model

for predicting responses to new motions depends on the type of training maneuver used

to estimate kernels. A model that includes Mach number effects significantly increases the

computational cost because it requires many calculations for each combination of angle of

attack and Mach number.


Figure 1.22 X–31 vortical flows using SARC–DDES turbulence model. The conditions are M∞ =0.18 and Re = 2× 106 , ’Ghoreyshi et al. (2013)’

Conclusions

The inadequacy and shortcomings of current industrial design procedures is apparent when

confronted with the stringent environmental constraints imposed to control the impact of

aviation on the environment and communities around airports. In need of fast turnaround

times, industrial procedures are based on low–fidelity empirical and linear approximations.

Having recognised the urgency to increase the realism of predictions for better designs and

reducing the time to bring new aircraft on the market, this Chapter presented state–of–the–

art methods for virtual flight simulation. The development of a unified approach, capable to


(a) Linear chirp (b) Spiral

Figure 1.23 Volterra reduced–order modeling using spiral and chirp training maneu-

vers, ’Ghoreyshi et al. (2013)’

(a) ROM based on linear responses (b) ROM based on nonlinear responses

Figure 1.24 Indicial functions reduced–order modeling using spiral and chirp training maneu-

vers, ’Ghoreyshi et al. (2013)’

assist the aircraft designer to investigate unconventional configurations and identify critical

design features that may jeopardise the aircraft programme if discovered late in the design

process, is instrumental to meet the ambitious goals of EU and US.


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